A Uniqueness Problem in Valued function Fields of Conics

Abstract

Let v0 be a valuation of a field K0 with value group G0. Let K be a function field of a conic over K0, and let v be an extension of v0 to K with value group G such that G/G0 is not a torsion group. Suppose that either (K0, v0) is henselian or v0 is of rank 1, the algebraic closure of K0 in K is a purely inseparable extension of K0, and G0 is a cofinal subset of G. In this paper, it is proved that there exists an explicitly constructible element t in K, with v(t) non-torsion modulo G0 such that the valuation of K0(t), obtained by restricting v, has a unique extension to K. This generalizes the result proved by Khanduja in the particular case, when K is a simple transcendental extension of K0 (compare [4]). The above result is an analogue of a result of Polzin proved for residually transcendental extensions [8].